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<div class="section" id="Introduction-to-atomman:-Gamma-surface-plotting">
<h1>Introduction to atomman: Gamma surface plotting<a class="headerlink" href="#Introduction-to-atomman:-Gamma-surface-plotting" title="Permalink to this headline">¶</a></h1>
<p><strong>Lucas M. Hale</strong>, <a class="reference external" href="mailto:lucas&#46;hale&#37;&#52;&#48;nist&#46;gov?Subject=ipr-demo">lucas<span>&#46;</span>hale<span>&#64;</span>nist<span>&#46;</span>gov</a>, <em>Materials Science and Engineering Division, NIST</em>.</p>
<p><a class="reference external" href="http://www.nist.gov/public_affairs/disclaimer.cfm">Disclaimers</a></p>
<div class="section" id="1.-Introduction">
<h2>1. Introduction<a class="headerlink" href="#1.-Introduction" title="Permalink to this headline">¶</a></h2>
<p>This Notebook outlines the GammaSurface class and the various stacking fault plots that it can generate. See the <a class="reference external" href="4.3._Stacking_fault_generator.html">4.3. Stacking fault generator Notebook</a> for generating systems containing stacking faults.</p>
<p><strong>Library Imports</strong></p>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[1]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="c1"># Standard Python libraries</span>
<span class="kn">from</span> <span class="nn">pathlib</span> <span class="k">import</span> <span class="n">Path</span>
<span class="kn">import</span> <span class="nn">datetime</span>

<span class="c1"># http://www.numpy.org/</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>

<span class="c1"># https://github.com/usnistgov/atomman</span>
<span class="kn">import</span> <span class="nn">atomman</span> <span class="k">as</span> <span class="nn">am</span>
<span class="kn">import</span> <span class="nn">atomman.unitconvert</span> <span class="k">as</span> <span class="nn">uc</span>

<span class="c1"># https://matplotlib.org/</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="o">%</span><span class="k">matplotlib</span> inline

<span class="c1"># Show atomman version</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;atomman version =&#39;</span><span class="p">,</span> <span class="n">am</span><span class="o">.</span><span class="n">__version__</span><span class="p">)</span>

<span class="c1"># Show date of Notebook execution</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Notebook executed on&#39;</span><span class="p">,</span> <span class="n">datetime</span><span class="o">.</span><span class="n">date</span><span class="o">.</span><span class="n">today</span><span class="p">())</span>
</pre></div>
</div>
</div>
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atomman version = 1.4.0
Notebook executed on 2021-08-05
</pre></div></div>
</div>
</div>
<div class="section" id="2.-Underlying-theory">
<h2>2. Underlying theory<a class="headerlink" href="#2.-Underlying-theory" title="Permalink to this headline">¶</a></h2>
<p>Generalized stacking fault energy maps (<span class="math notranslate nohighlight">\(\gamma\)</span>-surfaces) provide 2D maps of the stacking fault energy associated with arbitrary shifts along a crystallographic fault plane. Typically, the fault plane corresponds to one of the crystal’s slip planes and the stacking fault energies and pathways influence dislocation structures and dislocation slip.</p>
<p>A <span class="math notranslate nohighlight">\(\gamma\)</span>-surface is defined by the crystal and the fault plane. Starting in an unfaulted configuration, the atoms on one side of the fault plane are shifted by a vector <span class="math notranslate nohighlight">\(\mathbf{d}\)</span> which is within the fault plane. The generalized stacking fault energy for a given shift, <span class="math notranslate nohighlight">\(\gamma_{gsf}(\mathbf{d})\)</span>, is computed as the difference in potential energy, <span class="math notranslate nohighlight">\(U\)</span> of the shifted configuration relative to the corresponding unshifted configuration (<span class="math notranslate nohighlight">\(\mathbf{d}=\mathbf{0}\)</span>)
divided by the fault plane area, <span class="math notranslate nohighlight">\(A_{fault}\)</span></p>
<div class="math notranslate nohighlight">
\[\gamma_{gsf}(\mathbf{d}) = \frac{U(\mathbf{d}) - U(\mathbf{0})}{A_{fault}}\]</div>
<p>The measured <span class="math notranslate nohighlight">\(\gamma_{gsf}\)</span> values may be “unrelaxed”, in which case no atomic relaxations are allowed, or “relaxed”, in which atoms are allowed to move only in the direction normal to the slip plane to lower the system’s energy. For relaxed <span class="math notranslate nohighlight">\(\gamma\)</span>-surfaces, the relaxation distance, <span class="math notranslate nohighlight">\(\delta_{\bot}\)</span>, can be evaluated using the centers of mass of the atoms on the two sides of the fault plane, <span class="math notranslate nohighlight">\(\mathbf{R}^+\)</span> and <span class="math notranslate nohighlight">\(\mathbf{R}^-\)</span>, and measuring how the plane normal
component <span class="math notranslate nohighlight">\(\bot\)</span> changes relative to the unshifted state</p>
<div class="math notranslate nohighlight">
\[\mathbf{R} = \frac{\sum_{i=1}^{n}m_i\mathbf{r_i}}{\sum_{i=1}^{n}m_i}\]</div>
<div class="math notranslate nohighlight">
\[\delta_{\bot} = \left(R_{\bot}^+(\mathbf{d}) - R_{\bot}^-(\mathbf{d})\right) - \left(R_{\bot}^+(\mathbf{0}) - R_{\bot}^-(\mathbf{0})\right)\]</div>
<p>As a <span class="math notranslate nohighlight">\(\gamma\)</span>-surface is a 2D map, the applied shift, <span class="math notranslate nohighlight">\(\mathbf{d}\)</span>, can be expressed using two non-parallel vectors within the fault plane, <span class="math notranslate nohighlight">\(\mathbf{a_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{a_2}\)</span>. Taking <span class="math notranslate nohighlight">\(\mathbf{a_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{a_2}\)</span> as lattice vectors (i.e. vectors between two symetrically identical lattice sites) will define a 2D cell encompassing all possible unique stacking fault shifts for the fault plane. <span class="math notranslate nohighlight">\(\mathbf{d}\)</span> can then be represented using fractional
coordinates, <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span>, relative to <span class="math notranslate nohighlight">\(\mathbf{a_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{a_2}\)</span></p>
<div class="math notranslate nohighlight">
\[\mathbf{d} = a_1 \mathbf{a_1} + a_2 \mathbf{a_2}\]</div>
<p>The <span class="math notranslate nohighlight">\(\gamma_{gsf}\)</span> (and <span class="math notranslate nohighlight">\(\delta_{\bot}\)</span>) data is fit to interpolation models to provide continuous values between the measured points. Two interpolation models are fit to the data</p>
<ul class="simple">
<li><p>A multiquadric radial basis function interpolation.</p></li>
<li><p>A nearest-value interpolation.</p></li>
</ul>
<p>To perform the fits, the data is replicated across the periodic bounds by shifting the <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> values by <span class="math notranslate nohighlight">\(\pm 1\)</span>. The unique <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> values less than and closest to 0 and greater than and closest to 1 are identified, and all measured points between these two limits are fed into the fitting models to ensure smooth representation across the bounds. Any arbitrary shift values <span class="math notranslate nohighlight">\(\mathbf{d}\)</span> are converted into <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> values
between 0 and 1 for use in retrieving values from the fitted models.</p>
</div>
<div class="section" id="3.-Defining-gamma-surface-data">
<h2>3. Defining gamma surface data<a class="headerlink" href="#3.-Defining-gamma-surface-data" title="Permalink to this headline">¶</a></h2>
<p>The atomman.defect.GammaSurface class offers interpolation and plotting options for generalized stacking fault data.</p>
<div class="section" id="3.1.-Setting-values">
<h3>3.1. Setting values<a class="headerlink" href="#3.1.-Setting-values" title="Permalink to this headline">¶</a></h3>
<p>The stacking fault data can be set directly either during class initialization or by using the GammaSurface.set() method.</p>
<p>Parameters:</p>
<ul class="simple">
<li><p><strong>a1vect</strong> (<em>array-like object of length 3</em>) The a1 shifting vector. If box is given, a1vect is taken as a crystal lattice vector, otherwise as a Cartesian vector.</p></li>
<li><p><strong>a2vect</strong> (<em>array-like object of length 3</em>) The a2 shifting vector. If box is given, a1vect is taken as a crystal lattice vector, otherwise as a Cartesian vector.</p></li>
<li><p><strong>a1</strong> (<em>array-like object of length N</em>) List of fractional coordinates along a1vect corresponding to the E_gsf (and delta) values.</p></li>
<li><p><strong>a2</strong> (<em>array-like object of length N</em>) List of fractional coordinates along a2vect corresponding to the E_gsf (and delta) values.</p></li>
<li><p><strong>E_gsf</strong> (<em>array-like object of length N</em>) List of generalized stacking fault energies for the positions associated with the corresponding (a1, a2) fractional coordinates.</p></li>
<li><p><strong>box</strong> (<em>atomman.Box, optional</em>) Defines unit cell box dimensions for conversion between crystal lattice and Cartesian vectors. If not given, will be set as a square unit box, thus no conversion will occur (i.e. a1vect, a2vect will be Cartesian).</p></li>
<li><p><strong>delta</strong> (<em>array-like object of length N, optional</em>) List of change in displacements normal to the fault plane for the positions associated with the corresponding (a1, a2) fractional coordinates.</p></li>
</ul>
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<span></span><span class="c1"># Parameters for an Al fcc 111 stacking fault using 2008--Mendelev-M-I--Al--LAMMPS--ipr1</span>
<span class="n">box</span> <span class="o">=</span> <span class="n">am</span><span class="o">.</span><span class="n">Box</span><span class="o">.</span><span class="n">cubic</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="mf">4.045259794316263</span><span class="p">)</span>
<span class="n">a1vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">0.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
<span class="n">a2vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>

<span class="c1"># Measured 30x30 grid of generalized stacking fault data</span>
<span class="n">a1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> 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class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.06666666666666667</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.13333333333333333</span><span class="p">,</span> <span class="mf">0.16666666666666666</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.23333333333333334</span><span class="p">,</span> <span class="mf">0.26666666666666666</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3333333333333333</span><span class="p">,</span> <span class="mf">0.36666666666666664</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.43333333333333335</span><span class="p">,</span> <span class="mf">0.4666666666666667</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5333333333333333</span><span class="p">,</span> <span class="mf">0.5666666666666667</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.6333333333333333</span><span class="p">,</span> <span class="mf">0.6666666666666666</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">])</span>
<span class="n">a2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span class="mf">0.03333333333333333</span><span class="p">,</span> <span 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class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7333333333333333</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.7666666666666666</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8333333333333334</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.8666666666666667</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9333333333333333</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">,</span> <span class="mf">0.9666666666666667</span><span class="p">])</span>
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class="p">,</span> <span class="mf">0.0812011583611536</span><span class="p">,</span> <span class="mf">0.0854178570514108</span><span class="p">,</span> <span class="mf">0.097001190583768</span><span class="p">,</span> <span class="mf">0.11262352593976743</span><span class="p">,</span> <span class="mf">0.12996798226795647</span><span class="p">,</span> <span class="mf">0.1520827241580065</span><span class="p">,</span> <span class="mf">0.17737775568241432</span><span class="p">,</span> <span class="mf">0.19966471494916505</span><span class="p">,</span> <span class="mf">0.22334916339341504</span><span class="p">,</span> <span class="mf">0.24908491977631542</span><span class="p">,</span> <span class="mf">0.27617821394380115</span><span class="p">,</span> <span class="mf">0.3029593759366236</span><span class="p">,</span> <span class="mf">0.32630850753324125</span><span class="p">,</span> <span class="mf">0.34318151446927914</span><span class="p">,</span> <span class="mf">0.35294496236149087</span><span class="p">,</span> <span class="mf">0.35610344527063376</span><span class="p">,</span> <span class="mf">0.3529266892192737</span><span class="p">,</span> <span class="mf">0.34318151446928624</span><span class="p">,</span> <span class="mf">0.3263085075331986</span><span class="p">,</span> <span class="mf">0.3029593759362825</span><span class="p">,</span> <span class="mf">0.2761782139437159</span><span class="p">,</span> <span class="mf">0.2490849197762941</span><span class="p">,</span> <span class="mf">0.22334916339339372</span><span class="p">,</span> <span class="mf">0.19966471494916505</span><span class="p">,</span> <span class="mf">0.17737775568241432</span><span class="p">,</span> <span class="mf">0.1520827241580065</span><span class="p">,</span> <span class="mf">0.12996798226792805</span><span class="p">,</span> <span class="mf">0.08851254231704786</span><span class="p">,</span> <span class="mf">0.07078901159886186</span><span class="p">,</span> <span class="mf">0.060689274713794816</span><span class="p">,</span> <span class="mf">0.060689274713794816</span><span class="p">,</span> <span class="mf">0.07078901159886186</span><span class="p">,</span> <span class="mf">0.08851254231707628</span><span class="p">,</span> <span class="mf">0.10818940005012223</span><span class="p">,</span> <span class="mf">0.12986956796500237</span><span class="p">,</span> <span class="mf">0.15680287801056636</span><span class="p">,</span> <span class="mf">0.17955776385476696</span><span class="p">,</span> <span class="mf">0.20053351038428247</span><span class="p">,</span> <span class="mf">0.2233491633933795</span><span class="p">,</span> <span class="mf">0.2481491165990235</span><span class="p">,</span> <span class="mf">0.27401119238007254</span><span class="p">,</span> <span class="mf">0.2984966838195078</span><span class="p">,</span> <span class="mf">0.31818773979040316</span><span class="p">,</span> <span class="mf">0.331176612149477</span><span class="p">,</span> <span class="mf">0.3374568306061221</span><span class="p">,</span> <span class="mf">0.3374700402644564</span><span class="p">,</span> <span class="mf">0.3311766121494628</span><span class="p">,</span> <span class="mf">0.3181877397905808</span><span class="p">,</span> <span class="mf">0.29849668381974226</span><span class="p">,</span> <span class="mf">0.2740111923800441</span><span class="p">,</span> <span class="mf">0.2481491165990377</span><span class="p">,</span> <span class="mf">0.22334916339343636</span><span class="p">,</span> <span class="mf">0.20053351038428957</span><span class="p">,</span> <span class="mf">0.17955776385478828</span><span class="p">,</span> <span class="mf">0.15680287801056636</span><span class="p">,</span> <span class="mf">0.129869567965045</span><span class="p">,</span> <span class="mf">0.10818940005011513</span><span class="p">,</span> <span class="mf">0.06068821203136565</span><span class="p">,</span> <span class="mf">0.0456118143712132</span><span class="p">,</span> <span class="mf">0.04005475255581814</span><span class="p">,</span> <span class="mf">0.04561181437117057</span><span class="p">,</span> <span class="mf">0.06068821203162145</span><span class="p">,</span> <span class="mf">0.08036626481417386</span><span class="p">,</span> <span class="mf">0.10241054554465734</span><span class="p">,</span> <span class="mf">0.13017196439993484</span><span class="p">,</span> <span class="mf">0.15862805959559267</span><span class="p">,</span> <span class="mf">0.17955776385478828</span><span class="p">,</span> <span class="mf">0.19966471494916505</span><span class="p">,</span> <span class="mf">0.2218320536125873</span><span class="p">,</span> <span class="mf">0.24575740357897047</span><span class="p">,</span> <span class="mf">0.269738133292563</span><span class="p">,</span> <span class="mf">0.2910645412784305</span><span class="p">,</span> <span class="mf">0.3066522734923325</span><span class="p">,</span> <span class="mf">0.31581982531282904</span><span class="p">,</span> <span class="mf">0.31881098931053486</span><span class="p">,</span> <span class="mf">0.3158198253129001</span><span class="p">,</span> <span class="mf">0.3066522734923183</span><span class="p">,</span> <span class="mf">0.29106212506582096</span><span class="p">,</span> <span class="mf">0.2697381332918667</span><span class="p">,</span> <span class="mf">0.24575740357897757</span><span class="p">,</span> <span class="mf">0.22183205361280045</span><span class="p">,</span> <span class="mf">0.19966471494916505</span><span class="p">,</span> <span class="mf">0.17955776385477407</span><span class="p">,</span> <span class="mf">0.15862805959559978</span><span class="p">,</span> <span class="mf">0.13017196439993484</span><span class="p">,</span> <span class="mf">0.10241054554416706</span><span class="p">,</span> <span class="mf">0.08036626481417386</span><span class="p">,</span> <span class="mf">0.038158432534764586</span><span class="p">,</span> <span class="mf">0.02686634910992325</span><span class="p">,</span> <span class="mf">0.02686634910992325</span><span class="p">,</span> <span class="mf">0.03815843253483564</span><span class="p">,</span> <span class="mf">0.055965520644626565</span><span class="p">,</span> <span class="mf">0.07604824987024017</span><span class="p">,</span> <span class="mf">0.09954551522126565</span><span class="p">,</span> <span class="mf">0.13017196439993484</span><span class="p">,</span> <span class="mf">0.15680287801055925</span><span class="p">,</span> <span class="mf">0.17737775568241432</span><span class="p">,</span> <span class="mf">0.19748976356095227</span><span class="p">,</span> <span class="mf">0.21943396764225298</span><span class="p">,</span> <span class="mf">0.2422278736200667</span><span class="p">,</span> <span class="mf">0.26387327731772814</span><span class="p">,</span> <span class="mf">0.2812832166455621</span><span class="p">,</span> <span class="mf">0.2928239257262817</span><span class="p">,</span> <span class="mf">0.2984516331511742</span><span class="p">,</span> <span class="mf">0.2984451507659145</span><span class="p">,</span> <span class="mf">0.2928239257262675</span><span class="p">,</span> <span class="mf">0.2812561927627115</span><span class="p">,</span> <span class="mf">0.26387327731775656</span><span class="p">,</span> <span class="mf">0.24222787362003828</span><span class="p">,</span> <span class="mf">0.21943396764224588</span><span class="p">,</span> <span class="mf">0.19748976356096648</span><span class="p">,</span> <span class="mf">0.17737775568237168</span><span class="p">,</span> <span class="mf">0.15680287801059478</span><span class="p">,</span> <span class="mf">0.13017196439994194</span><span class="p">,</span> <span class="mf">0.09954551522132249</span><span class="p">,</span> <span class="mf">0.07604824987022596</span><span class="p">,</span> <span class="mf">0.055965520644626565</span><span class="p">,</span> <span class="mf">0.019436630015306378</span><span class="p">,</span> <span class="mf">0.013315364007297603</span><span class="p">,</span> <span class="mf">0.019436630015306378</span><span class="p">,</span> <span class="mf">0.03496863398257943</span><span class="p">,</span> <span class="mf">0.05455642334713673</span><span class="p">,</span> <span class="mf">0.07604824987024017</span><span class="p">,</span> <span class="mf">0.10241054554417417</span><span class="p">,</span> <span class="mf">0.12986956796501659</span><span class="p">,</span> <span class="mf">0.1520827241580207</span><span class="p">,</span> <span class="mf">0.17278323635773063</span><span class="p">,</span> <span class="mf">0.1938092073899611</span><span class="p">,</span> <span class="mf">0.2158073704211887</span><span class="p">,</span> <span class="mf">0.23737736077522698</span><span class="p">,</span> <span class="mf">0.25604565034051063</span><span class="p">,</span> <span class="mf">0.26952461106114356</span><span class="p">,</span> <span class="mf">0.27741840435356835</span><span class="p">,</span> <span class="mf">0.2799730567250549</span><span class="p">,</span> <span class="mf">0.27741840435355414</span><span class="p">,</span> <span class="mf">0.26952461106114356</span><span class="p">,</span> <span class="mf">0.25604565034053905</span><span class="p">,</span> <span class="mf">0.23737736077521987</span><span class="p">,</span> <span class="mf">0.21580737042112474</span><span class="p">,</span> <span class="mf">0.1938092073899682</span><span class="p">,</span> <span class="mf">0.17278323636047332</span><span class="p">,</span> <span class="mf">0.1520827241580065</span><span class="p">,</span> <span class="mf">0.1298695679650237</span><span class="p">,</span> <span class="mf">0.10241054554415996</span><span class="p">,</span> <span class="mf">0.07604824987025438</span><span class="p">,</span> <span class="mf">0.05455642334713673</span><span class="p">,</span> <span class="mf">0.03496863398257233</span><span class="p">,</span> <span class="mf">0.005541623615314961</span><span class="p">,</span> <span class="mf">0.005541623615314961</span><span class="p">,</span> <span class="mf">0.016933037985893407</span><span class="p">,</span> <span class="mf">0.03496863398256522</span><span class="p">,</span> <span class="mf">0.05596552064464788</span><span class="p">,</span> <span class="mf">0.08036626481413833</span><span class="p">,</span> <span class="mf">0.10818940005012223</span><span class="p">,</span> <span class="mf">0.12996798226794226</span><span class="p">,</span> <span class="mf">0.14803652963331615</span><span class="p">,</span> <span class="mf">0.1660858553276796</span><span class="p">,</span> <span class="mf">0.187787002826461</span><span class="p">,</span> <span class="mf">0.21056922427537472</span><span class="p">,</span> <span class="mf">0.23091343202591474</span><span class="p">,</span> <span class="mf">0.24645629091584453</span><span class="p">,</span> <span class="mf">0.2563945023545955</span><span class="p">,</span> <span class="mf">0.2611668393310964</span><span class="p">,</span> <span class="mf">0.2611668393310964</span><span class="p">,</span> <span class="mf">0.2563945023546097</span><span class="p">,</span> <span class="mf">0.24645629091583032</span><span class="p">,</span> <span class="mf">0.23091343202594317</span><span class="p">,</span> <span class="mf">0.21056922427537472</span><span class="p">,</span> <span class="mf">0.1877870028264681</span><span class="p">,</span> <span class="mf">0.16608585532770803</span><span class="p">,</span> <span class="mf">0.14803652963334457</span><span class="p">,</span> <span class="mf">0.12996798226791384</span><span class="p">,</span> <span class="mf">0.10818940005012223</span><span class="p">,</span> <span class="mf">0.08036626481415965</span><span class="p">,</span> <span class="mf">0.055965520644612354</span><span class="p">,</span> <span class="mf">0.03496863398257233</span><span class="p">,</span> <span class="mf">0.016933037985893407</span><span class="p">])</span>

<span class="n">gamma</span> <span class="o">=</span> <span class="n">am</span><span class="o">.</span><span class="n">defect</span><span class="o">.</span><span class="n">GammaSurface</span><span class="p">(</span><span class="n">a1vect</span><span class="o">=</span><span class="n">a1vect</span><span class="p">,</span> <span class="n">a2vect</span><span class="o">=</span><span class="n">a2vect</span><span class="p">,</span> <span class="n">a1</span><span class="o">=</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="o">=</span><span class="n">a2</span><span class="p">,</span>
                               <span class="n">E_gsf</span><span class="o">=</span><span class="n">E_gsf</span><span class="p">,</span> <span class="n">box</span><span class="o">=</span><span class="n">box</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="n">delta</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="3.2.-Class-attributes">
<h3>3.2. Class attributes<a class="headerlink" href="#3.2.-Class-attributes" title="Permalink to this headline">¶</a></h3>
<p>The GammaSurface class has the following attributes for the data</p>
<ul class="simple">
<li><p><strong>a1vect, a2vect</strong> the two shifting vectors as given.</p></li>
<li><p><strong>box</strong> the unit cell box as given.</p></li>
<li><p><strong>planenormal</strong> a Cartesian vector normal to the fault plane.</p></li>
<li><p><strong>data</strong> a pandas.DataFrame tabulating the measured stacking fault data. The data columns are</p>
<ul>
<li><p><strong>a1</strong> fractional coordinates along the a1vect.</p></li>
<li><p><strong>a2</strong> fractional coordinates along the a2vect.</p></li>
<li><p><strong>E_gsf</strong> generalized stacking fault energies measured for each (a1, a2) fault shift.</p></li>
<li><p><strong>delta</strong> (optional) relaxation distances measured for each (a1, a2) fault shift.</p></li>
</ul>
</li>
</ul>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[3]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;gamma.a1vect -&gt;&#39;</span><span class="p">,</span> <span class="n">gamma</span><span class="o">.</span><span class="n">a1vect</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;gamma.a2vect -&gt;&#39;</span><span class="p">,</span> <span class="n">gamma</span><span class="o">.</span><span class="n">a2vect</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;gamma.planenormal -&gt;&#39;</span><span class="p">,</span> <span class="n">gamma</span><span class="o">.</span><span class="n">planenormal</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;gamma.box -&gt;&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">gamma</span><span class="o">.</span><span class="n">box</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="nboutput nblast docutils container">
<div class="prompt empty docutils container">
</div>
<div class="output_area docutils container">
<div class="highlight"><pre>
gamma.a1vect -&gt; [ 0.  -0.5  0.5]
gamma.a2vect -&gt; [ 0.5 -0.5  0. ]
gamma.planenormal -&gt; [0.57735027 0.57735027 0.57735027]
gamma.box -&gt;
avect =  [ 4.045,  0.000,  0.000]
bvect =  [ 0.000,  4.045,  0.000]
cvect =  [ 0.000,  0.000,  4.045]
origin = [ 0.000,  0.000,  0.000]
</pre></div></div>
</div>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[4]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="n">gamma</span><span class="o">.</span><span class="n">data</span>
</pre></div>
</div>
</div>
<div class="nboutput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[4]:
</pre></div>
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<div class="output_area rendered_html docutils container">
<div>
<style scoped>
    .dataframe tbody tr th:only-of-type {
        vertical-align: middle;
    }

    .dataframe tbody tr th {
        vertical-align: top;
    }

    .dataframe thead th {
        text-align: right;
    }
</style>
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>a1</th>
      <th>a2</th>
      <th>E_gsf</th>
      <th>delta</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>0</td>
      <td>0.000000</td>
      <td>0.000000</td>
      <td>0.000000</td>
      <td>0.000000</td>
    </tr>
    <tr>
      <td>1</td>
      <td>0.033333</td>
      <td>0.000000</td>
      <td>0.000328</td>
      <td>0.005542</td>
    </tr>
    <tr>
      <td>2</td>
      <td>0.066667</td>
      <td>0.000000</td>
      <td>0.001320</td>
      <td>0.019437</td>
    </tr>
    <tr>
      <td>3</td>
      <td>0.100000</td>
      <td>0.000000</td>
      <td>0.003008</td>
      <td>0.038158</td>
    </tr>
    <tr>
      <td>4</td>
      <td>0.133333</td>
      <td>0.000000</td>
      <td>0.005299</td>
      <td>0.060688</td>
    </tr>
    <tr>
      <td>...</td>
      <td>...</td>
      <td>...</td>
      <td>...</td>
      <td>...</td>
    </tr>
    <tr>
      <td>895</td>
      <td>0.833333</td>
      <td>0.966667</td>
      <td>0.009792</td>
      <td>0.108189</td>
    </tr>
    <tr>
      <td>896</td>
      <td>0.866667</td>
      <td>0.966667</td>
      <td>0.007090</td>
      <td>0.080366</td>
    </tr>
    <tr>
      <td>897</td>
      <td>0.900000</td>
      <td>0.966667</td>
      <td>0.004494</td>
      <td>0.055966</td>
    </tr>
    <tr>
      <td>898</td>
      <td>0.933333</td>
      <td>0.966667</td>
      <td>0.002400</td>
      <td>0.034969</td>
    </tr>
    <tr>
      <td>899</td>
      <td>0.966667</td>
      <td>0.966667</td>
      <td>0.001009</td>
      <td>0.016933</td>
    </tr>
  </tbody>
</table>
<p>900 rows × 4 columns</p>
</div></div>
</div>
</div>
<div class="section" id="3.3.-Saving-and-loading-data-model">
<h3>3.3. Saving and loading data model<a class="headerlink" href="#3.3.-Saving-and-loading-data-model" title="Permalink to this headline">¶</a></h3>
<p>The GammaSurface.model() method allows for the saving and loading of a GammaSurface object’s data as a JSON/XML data model.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>model</strong> (<em>str, file-like object or DataModelDict, optional</em>) XML/JSON content to extract gamma surface energy from. If not given, model content will be generated.</p></li>
<li><p><strong>length_unit</strong> (<em>str, optional</em>) Units to report delta displacement values in when a new model is generated. Default value is ‘angstrom’.</p></li>
<li><p><strong>energyperarea_unit</strong> (<em>str, optional</em>) Units to report fault energy values in when a new model is generated. Default value is ‘mJ/m^2’.</p></li>
</ul>
<p>Returns</p>
<p>(<em>DataModelDict</em>) A data model containing the stacking fault data of the GammaSurface object. Returned if model is not given.</p>
<div class="section" id="3.3.1.-Save-data-model">
<h4>3.3.1. Save data model<a class="headerlink" href="#3.3.1.-Save-data-model" title="Permalink to this headline">¶</a></h4>
<p>Calling GammaSurface.model() without supplying the ‘model’ parameter will return a data model of the content.</p>
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<span></span><span class="n">model</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">model</span><span class="p">()</span>
<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s1">&#39;gamma.json&#39;</span><span class="p">,</span> <span class="s1">&#39;w&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
    <span class="n">model</span><span class="o">.</span><span class="n">json</span><span class="p">(</span><span class="n">fp</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
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</div>
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</div>
<div class="section" id="3.3.2.-Load-data-model">
<h4>3.3.2. Load data model<a class="headerlink" href="#3.3.2.-Load-data-model" title="Permalink to this headline">¶</a></h4>
<p>The saved data can then be loaded by using the ‘model’ parameter when initializing a GammaSurface, or with the GammaSurface.model() method.</p>
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<span></span><span class="n">gamma</span> <span class="o">=</span> <span class="n">am</span><span class="o">.</span><span class="n">defect</span><span class="o">.</span><span class="n">GammaSurface</span><span class="p">(</span><span class="n">model</span><span class="o">=</span><span class="s1">&#39;gamma.json&#39;</span><span class="p">)</span>
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</div>
</div>
</div>
</div>
</div>
<div class="section" id="4.-Plotting-methods">
<h2>4. Plotting methods<a class="headerlink" href="#4.-Plotting-methods" title="Permalink to this headline">¶</a></h2>
<div class="section" id="4.1.-E_gsf_line_plot()">
<h3>4.1. <strong>E_gsf_line_plot()</strong><a class="headerlink" href="#4.1.-E_gsf_line_plot()" title="Permalink to this headline">¶</a></h3>
<p>Generates a line plot for the interpolated generalized stacking fault energy along a specified vector in the (a1, a2) plane.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>vect</strong> (<em>numpy.array, optional</em>) Vector to plot the gsf along. If box is set, this vect will be a lattice vector, otherwise it will be a Cartesian vector. Must be in the plane defined by the GammaSurface object’s a1vect and a2vect vectors. Default value will use the set a1vect.</p></li>
<li><p><strong>num</strong> (<em>int, optional</em>) The number of points to evaluate the generalized stacking fault energy for. Default value is 100 if smooth is True, otherwise is number of unique a1 values from 0 to 1.</p></li>
<li><p><strong>smooth</strong> (<em>bool, optional</em>) If True (default), then plot shows smooth interpolated values. If False, plot shows nearest raw data values.</p></li>
<li><p><strong>length_unit</strong> (<em>str, optional</em>) The unit of length to display the x-axis coordinates in. Default value is ‘Å’.</p></li>
<li><p><strong>energyperarea_unit</strong> (<em>str, optional</em>) The unit of energy per area to display the stacking fault energies in. Default value is ‘eV/Å^2’.</p></li>
<li><p><strong>figsize</strong> (<em>tuple, optional</em>) The x,y size of the figure to return. Default value is (10, 6).</p></li>
<li><p><strong>fig</strong> (<em>matplotlib.figure, optional</em>) An existing figure object to add the new plot to. If not given, a new figure is generated.</p></li>
<li><p><strong>**kwargs</strong> (<em>dict, optional</em>) Additional keywords are passed into the underlying matplotlib.pyplot.plot(). This allows control of such things like line color, style, etc.</p></li>
</ul>
<p>Returns</p>
<p>(<em>matplotlib.figure</em>)</p>
<p>If vect is not given, will use a1vect</p>
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<span></span><span class="c1"># Generate RBF interpolated plot</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_line_plot</span><span class="p">(</span><span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;RBF&#39;</span><span class="p">)</span>

<span class="c1"># Generate measured (nearest) plot and append to above</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_line_plot</span><span class="p">(</span><span class="n">smooth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k:o&#39;</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="n">fig</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;measured&#39;</span><span class="p">)</span>

<span class="c1"># Additional plot manipulations</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="s1">&#39;x-large&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>The important direction for fcc {111} stacking faults is a/2&lt;112&gt;, and energy values in mJ/m^2</p>
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<span></span><span class="c1"># Generate RBF interpolated plot</span>
<span class="n">vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_line_plot</span><span class="p">(</span><span class="n">vect</span><span class="o">=</span><span class="n">vect</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k-&#39;</span><span class="p">,</span> <span class="n">energyperarea_unit</span><span class="o">=</span><span class="s1">&#39;mJ/m^2&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;RBF&#39;</span><span class="p">)</span>

<span class="c1"># Generate measured (nearest) plot and append to above</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_line_plot</span><span class="p">(</span><span class="n">vect</span><span class="o">=</span><span class="n">vect</span><span class="p">,</span> <span class="n">smooth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k:o&#39;</span><span class="p">,</span>
                            <span class="n">energyperarea_unit</span><span class="o">=</span><span class="s1">&#39;mJ/m^2&#39;</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="n">fig</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;measured&#39;</span><span class="p">)</span>

<span class="c1"># Additional plot manipulations</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="s1">&#39;x-large&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="4.2.-delta_line_plot()">
<h3>4.2. <strong>delta_line_plot()</strong><a class="headerlink" href="#4.2.-delta_line_plot()" title="Permalink to this headline">¶</a></h3>
<p>Generates a line plot for the interpolated relaxation distances along a specified vector in the (a1, a2) plane.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>vect</strong> (<em>numpy.array, optional</em>) Vector to plot the gsf along. If box is set, this vect will be a lattice vector, otherwise it will be a Cartesian vector. Must be in the plane defined by the GammaSurface object’s a1vect and a2vect vectors. Default value will use the set a1vect.</p></li>
<li><p><strong>num</strong> (<em>int, optional</em>) The number of points to evaluate the generalized stacking fault energy for. Default value is 100 if smooth is True, otherwise is number of unique a1 values from 0 to 1.</p></li>
<li><p><strong>smooth</strong> (<em>bool, optional</em>) If True (default), then plot shows smooth interpolated values. If False, plot shows nearest raw data values.</p></li>
<li><p><strong>length_unit</strong> (<em>str, optional</em>) The unit of length to display the x-axis coordinates in. Default value is ‘Å’.</p></li>
<li><p><strong>figsize</strong> (<em>tuple, optional</em>) The x,y size of the figure to return. Default value is (10, 6).</p></li>
<li><p><strong>fig</strong> (<em>matplotlib.figure, optional</em>) An existing figure object to add the new plot to. If not given, a new figure is generated.</p></li>
<li><p><strong>**kwargs</strong> (<em>dict, optional</em>) Additional keywords are passed into the underlying matplotlib.pyplot.plot(). This allows control of such things like line color, style, etc.</p></li>
</ul>
<p>Returns</p>
<p>(<em>matplotlib.figure</em>)</p>
<p>If vect is not given, will use a1vect</p>
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<span></span><span class="c1"># Generate RBF interpolated plot</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_line_plot</span><span class="p">(</span><span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;RBF&#39;</span><span class="p">)</span>

<span class="c1"># Generate measured (nearest) plot and append to above</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_line_plot</span><span class="p">(</span><span class="n">smooth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k:o&#39;</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="n">fig</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;measured&#39;</span><span class="p">)</span>

<span class="c1"># Additional plot manipulations</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="s1">&#39;x-large&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>The important direction for fcc {111} stacking faults is a/2&lt;112&gt;</p>
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<span></span><span class="c1"># Generate RBF interpolated plot</span>
<span class="n">vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_line_plot</span><span class="p">(</span><span class="n">vect</span><span class="o">=</span><span class="n">vect</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;RBF&#39;</span><span class="p">)</span>

<span class="c1"># Generate measured (nearest) plot and append to above</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_line_plot</span><span class="p">(</span><span class="n">vect</span><span class="o">=</span><span class="n">vect</span><span class="p">,</span> <span class="n">smooth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;k:o&#39;</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="n">fig</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;measured&#39;</span><span class="p">)</span>

<span class="c1"># Additional plot manipulations</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="s1">&#39;x-large&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="4.3.-E_gsf_surface_plot()">
<h3>4.3. <strong>E_gsf_surface_plot()</strong><a class="headerlink" href="#4.3.-E_gsf_surface_plot()" title="Permalink to this headline">¶</a></h3>
<p>Creates a 2D surface plot from the stacking fault energy values.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>normalize</strong> (<em>bool, optional</em>) Flag indicating if axes are Cartesian (False, default) or normalized by a1, a2 vectors (True).</p></li>
<li><p><strong>smooth</strong> (<em>bool, optional</em>) If True (default), then plot shows smooth interpolated values. If False, plot shows nearest raw data values.</p></li>
<li><p><strong>a1vect</strong> (<em>np.array, optional</em>) Crystal vector for the a1 vector to use for plotting. Default value of None uses the saved a1vect.</p></li>
<li><p><strong>a2vect</strong> (<em>np.array, optional</em>) Crystal vector for the a2 vector to use for plotting. Default value of None uses the saved a2vect.</p></li>
<li><p><strong>xvect</strong> (<em>numpy.array, optional</em>) Crystal vector to align with the plotting x-axis for non-normalized plots. If not given, this is taken as the Cartesian of a1vect.</p></li>
<li><p><strong>length_unit</strong> (<em>str, optional</em>) The unit of length to display non-normalized axes values in. Default value is ‘Å’.</p></li>
<li><p><strong>energyperarea_unit</strong> (<em>str, optional</em>) The unit of energy per area to display the stacking fault energies in. Default value is ‘eV/Å^2’.</p></li>
<li><p><strong>numx</strong> (<em>int, optional</em>) The number of plotting points to use along the x-axis. Default value is 100.</p></li>
<li><p><strong>numy</strong> (<em>int, optional</em>) The number of plotting points to use along the y-axis. Default value is 100.</p></li>
<li><p><strong>figsize</strong> (<em>tuple or None, optional</em>) The figure’s x,y dimensions. If None (default), the values are scaled such that the x,y spacings are approximately equal, and the larger of the two values is set to 10.</p></li>
<li><p><strong>**kwargs</strong> (<em>dict, optional</em>) Additional keywords are passed into the underlying matplotlib.pyplot.pcolormesh(). This allows control of such things like the colormap (cmap).</p></li>
</ul>
<p>Returns</p>
<p>(<em>matplotlib.figure</em>)</p>
<p>Default setting</p>
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<span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_surface_plot</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Plotting actual measured values</p>
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<span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_surface_plot</span><span class="p">(</span><span class="n">smooth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<p>Specifying a different a1vect and a2vect to use for plotting</p>
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<span></span><span class="n">plot_a1vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>
<span class="n">plot_a2vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>

<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">E_gsf_surface_plot</span><span class="p">(</span><span class="n">a1vect</span><span class="o">=</span><span class="n">plot_a1vect</span><span class="p">,</span> <span class="n">a2vect</span><span class="o">=</span><span class="n">plot_a2vect</span><span class="p">,</span> <span class="n">energyperarea_unit</span><span class="o">=</span><span class="s1">&#39;mJ/m^2&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;jet&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="4.4.-delta_surface_plot()">
<h3>4.4. <strong>delta_surface_plot()</strong><a class="headerlink" href="#4.4.-delta_surface_plot()" title="Permalink to this headline">¶</a></h3>
<p>Creates a 2D surface plot from the relaxation distance values.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>normalize</strong> (<em>bool, optional</em>) Flag indicating if axes are Cartesian (False, default) or normalized by a1, a2 vectors (True).</p></li>
<li><p><strong>smooth</strong> (<em>bool, optional</em>) If True (default), then plot shows smooth interpolated values. If False, plot shows nearest raw data values.</p></li>
<li><p><strong>a1vect</strong> (<em>np.array, optional</em>) Crystal vector for the a1 vector to use for plotting. Default value of None uses the saved a1vect.</p></li>
<li><p><strong>a2vect</strong> (<em>np.array, optional</em>) Crystal vector for the a2 vector to use for plotting. Default value of None uses the saved a2vect.</p></li>
<li><p><strong>xvect</strong> (<em>numpy.array, optional</em>) Crystal vector to align with the plotting x-axis for non-normalized plots. If not given, this is taken as the Cartesian of a1vect.</p></li>
<li><p><strong>length_unit</strong> (<em>str, optional</em>) The unit of length to display non-normalized axes values in. Default value is ‘Å’.</p></li>
<li><p><strong>numx</strong> (<em>int, optional</em>) The number of plotting points to use along the x-axis. Default value is 100.</p></li>
<li><p><strong>numy</strong> (<em>int, optional</em>) The number of plotting points to use along the y-axis. Default value is 100.</p></li>
<li><p><strong>figsize</strong> (<em>tuple or None, optional</em>) The figure’s x,y dimensions. If None (default), the values are scaled such that the x,y spacings are approximately equal, and the larger of the two values is set to 10.</p></li>
<li><p><strong>**kwargs</strong> (<em>dict, optional</em>) Additional keywords are passed into the underlying matplotlib.pyplot.pcolormesh(). This allows control of such things like the colormap (cmap).</p></li>
</ul>
<p>Returns</p>
<p>(<em>matplotlib.figure</em>)</p>
<p>Default setting</p>
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<span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_surface_plot</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Plotting actual measured values</p>
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<span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_surface_plot</span><span class="p">(</span><span class="n">smooth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Specifying a different a1vect and a2vect to use for plotting</p>
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<span></span><span class="n">plot_a1vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>
<span class="n">plot_a2vect</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>

<span class="n">fig</span> <span class="o">=</span> <span class="n">gamma</span><span class="o">.</span><span class="n">delta_surface_plot</span><span class="p">(</span><span class="n">a1vect</span><span class="o">=</span><span class="n">plot_a1vect</span><span class="p">,</span> <span class="n">a2vect</span><span class="o">=</span><span class="n">plot_a2vect</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;jet&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="5.-Minimum-energy-slip-path">
<h2>5. Minimum energy slip path<a class="headerlink" href="#5.-Minimum-energy-slip-path" title="Permalink to this headline">¶</a></h2>
<p><em>Added version 1.3.7</em></p>
<p>The path() and build_path() methods allow for the construction of atomman.mep.Path objects where the energy function associated with the path coordinates is automatically set using the generalized stacking fault energy. path() allows for all 2D coordinates along the path to be specified, while build_path() allows for a path to be constructed based only on the endpoints and optionally one intermediate junction point.</p>
<p><strong>Documentation coming soon…</strong></p>
</div>
<div class="section" id="6.-Utility-methods">
<h2>6. Utility methods<a class="headerlink" href="#6.-Utility-methods" title="Permalink to this headline">¶</a></h2>
<p>The remaining class methods are utility methods for generating the plots. Many are focused on transforming between different coordinate systems:</p>
<ol class="arabic simple">
<li><p>(a1, a2) : fractional coordinates along a1vect and a2vect.</p></li>
<li><p>pos : Cartesian position coordinates.</p></li>
<li><p>(x, y) : plotting coordinates.</p></li>
</ol>
<p>Transforming from (a1, a2) to pos requires a1vect, a2vect, and optionally the box. Transforming from pos to (x, y) requires specifying which Cartesian vector to align with the x-axis of the plot.</p>
<div class="section" id="6.1.-E_gsf()">
<h3>6.1. E_gsf()<a class="headerlink" href="#6.1.-E_gsf()" title="Permalink to this headline">¶</a></h3>
<p>Returns values for generalized stacking fault energy interpolated from the raw data. Values can be obtained relative to a1, a2 fractional coordinates, x, y plotting coordinates, or pos Cartesian coordinates.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>a1</strong> (<em>float(s), optional</em>) Fractional coordinate(s) along a1vect.</p></li>
<li><p><strong>a2</strong> (<em>float(s), optional</em>) Fractional coordinate(s) along a2vect.</p></li>
<li><p><strong>pos</strong> (<em>np.array, optional</em>) 3D Cartesian position vector(s).</p></li>
<li><p><strong>x</strong> (<em>float(s), optional</em>) Plotting x coordinate(s).</p></li>
<li><p><strong>y</strong> (<em>float(s), optional</em>) Plotting y coordinate(s).</p></li>
<li><p><strong>a1vect</strong> (<em>np.array, optional</em>) Vector for the a1 fractional coordinates. Default value of None uses the saved a1vect.</p></li>
<li><p><strong>a2vect</strong> (<em>np.array, optional</em>) Vector for the a2 fractional coordinates. Default value of None uses the saved a2vect.</p></li>
<li><p><strong>xvect</strong> (<em>np.array, optional</em>) Cartesian vector corresponding to the plotting x-axis. If None (default), this is taken as the Cartesian of a1vect.</p></li>
<li><p><strong>smooth</strong> (<em>bool, optional</em>) If True (default) the returned values are smoothed using a RBF fit. If False, the closest measured values are returned.</p></li>
</ul>
<p>Returns</p>
<ul class="simple">
<li><p>(<em>float or numpy.ndarray</em>) the energy values.</p></li>
</ul>
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<span></span><span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;Close to E_isf = {uc.get_in_units(gamma.E_gsf(a1=1/3, a2=1/3), &quot;mJ/m^2&quot;)} mJ/m^2&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;Close to E_usf = {uc.get_in_units(gamma.E_gsf(a1=2/3, a2=2/3), &quot;mJ/m^2&quot;)} mJ/m^2&#39;</span><span class="p">)</span>
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Close to E_isf = 127.0843667373251 mJ/m^2
Close to E_usf = 471.0258819751663 mJ/m^2
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<div class="section" id="6.2-delta()">
<h3>6.2 delta()<a class="headerlink" href="#6.2-delta()" title="Permalink to this headline">¶</a></h3>
<p>Returns values for the relaxation distance interpolated from the raw data. Values can be obtained relative to a1, a2 fractional coordinates, x, y plotting coordinates, or pos Cartesian coordinates.</p>
<p>Parameters</p>
<ul class="simple">
<li><p><strong>a1</strong> (<em>float(s), optional</em>) Fractional coordinate(s) along a1vect.</p></li>
<li><p><strong>a2</strong> (<em>float(s), optional</em>) Fractional coordinate(s) along a2vect.</p></li>
<li><p><strong>pos</strong> (<em>np.array, optional</em>) 3D Cartesian position vector(s).</p></li>
<li><p><strong>x</strong> (<em>float(s), optional</em>) Plotting x coordinate(s).</p></li>
<li><p><strong>y</strong> (<em>float(s), optional</em>) Plotting y coordinate(s).</p></li>
<li><p><strong>a1vect</strong> (<em>np.array, optional</em>) Vector for the a1 fractional coordinates. Default value of None uses the saved a1vect.</p></li>
<li><p><strong>a2vect</strong> (<em>np.array, optional</em>) Vector for the a2 fractional coordinates. Default value of None uses the saved a2vect.</p></li>
<li><p><strong>xvect</strong> (<em>np.array, optional</em>) Cartesian vector corresponding to the plotting x-axis. If None (default), this is taken as the Cartesian of a1vect.</p></li>
<li><p><strong>smooth</strong> (<em>bool, optional</em>) If True (default) the returned values are smoothed using a RBF fit. If False, the closest measured values are returned.</p></li>
</ul>
<p>Returns</p>
<ul class="simple">
<li><p>(<em>float or numpy.ndarray</em>) the relaxation distance values.</p></li>
</ul>
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<span></span><span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;Close to δ_isf = {uc.get_in_units(gamma.delta(a1=1/3, a2=1/3), &quot;angstrom&quot;)} angstrom&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;Close to δ_usf = {uc.get_in_units(gamma.delta(a1=2/3, a2=2/3), &quot;angstrom&quot;)} angstrom&#39;</span><span class="p">)</span>
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Close to δ_isf = 0.04908088528014787 angstrom
Close to δ_usf = 0.3865334761953485 angstrom
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<div class="section" id="6.3-Conversion-methods">
<h3>6.3 Conversion methods<a class="headerlink" href="#6.3-Conversion-methods" title="Permalink to this headline">¶</a></h3>
<p>Parameters as defined above</p>
<ul class="simple">
<li><p><strong>a12_to_pos(a1, a2, a1vect=None, a2vect=None)</strong></p></li>
<li><p><strong>pos_to_xy(pos, xvect=None)</strong></p></li>
<li><p><strong>a12_to_xy(a1, a2, a1vect=None, a2vect=None, xvect=None)</strong></p></li>
<li><p><strong>pos_to_a12(pos, a1vect=None, a2vect=None)</strong></p></li>
<li><p><strong>xy_to_pos(x, y, xvect=None)</strong></p></li>
<li><p><strong>xy_to_a12(x, y, a1vect=None, a2vect=None, xvect=None)</strong></p></li>
</ul>
<p><strong>File cleanup</strong></p>
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<span></span><span class="n">Path</span><span class="p">(</span><span class="s1">&#39;gamma.json&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">unlink</span><span class="p">()</span>
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  <h3><a href="../index.html">Table of Contents</a></h3>
  <ul>
<li><a class="reference internal" href="#">Introduction to atomman: Gamma surface plotting</a><ul>
<li><a class="reference internal" href="#1.-Introduction">1. Introduction</a></li>
<li><a class="reference internal" href="#2.-Underlying-theory">2. Underlying theory</a></li>
<li><a class="reference internal" href="#3.-Defining-gamma-surface-data">3. Defining gamma surface data</a><ul>
<li><a class="reference internal" href="#3.1.-Setting-values">3.1. Setting values</a></li>
<li><a class="reference internal" href="#3.2.-Class-attributes">3.2. Class attributes</a></li>
<li><a class="reference internal" href="#3.3.-Saving-and-loading-data-model">3.3. Saving and loading data model</a><ul>
<li><a class="reference internal" href="#3.3.1.-Save-data-model">3.3.1. Save data model</a></li>
<li><a class="reference internal" href="#3.3.2.-Load-data-model">3.3.2. Load data model</a></li>
</ul>
</li>
</ul>
</li>
<li><a class="reference internal" href="#4.-Plotting-methods">4. Plotting methods</a><ul>
<li><a class="reference internal" href="#4.1.-E_gsf_line_plot()">4.1. <strong>E_gsf_line_plot()</strong></a></li>
<li><a class="reference internal" href="#4.2.-delta_line_plot()">4.2. <strong>delta_line_plot()</strong></a></li>
<li><a class="reference internal" href="#4.3.-E_gsf_surface_plot()">4.3. <strong>E_gsf_surface_plot()</strong></a></li>
<li><a class="reference internal" href="#4.4.-delta_surface_plot()">4.4. <strong>delta_surface_plot()</strong></a></li>
</ul>
</li>
<li><a class="reference internal" href="#5.-Minimum-energy-slip-path">5. Minimum energy slip path</a></li>
<li><a class="reference internal" href="#6.-Utility-methods">6. Utility methods</a><ul>
<li><a class="reference internal" href="#6.1.-E_gsf()">6.1. E_gsf()</a></li>
<li><a class="reference internal" href="#6.2-delta()">6.2 delta()</a></li>
<li><a class="reference internal" href="#6.3-Conversion-methods">6.3 Conversion methods</a></li>
</ul>
</li>
</ul>
</li>
</ul>

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